ΘΕΜΑΤΑ
1. In a triangle 
the points 
are the midpoints of the segments 
respectively. If 
is the point of intersection of 
and 
, and 
is the point of intersection of 
and , prove that:
the points are the midpoints of the segments respectively. If is the point of intersection of and , and is the point of intersection of and , prove that:(a) 
(b) 
(c) 
(d) the area of is 
times that of 
.
2.Let be an acute triangle, 
its altitudes and 
its orthocenter. Let 
be the internal and external bisectors of angle 
. Let 
be the midpoints of 
, respectively. Prove that:
(a)
is perpendicular to 
(b) if cuts the segments at the points 
, then 
.
3. Prove that among
natural numbers whose prime divisors are in the set 
there exist four numbers whose product is the fourth power of an integer.
is times that of . be an acute triangle, its altitudes and its orthocenter. Let be the internal and external bisectors of angle . Let be the midpoints of , respectively. Prove that:(a)
is perpendicular to (b) if
cuts the segments at the points , then .3. Prove that among
natural numbers whose prime divisors are in the set there exist four numbers whose product is the fourth power of an integer.
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